On the concept of ``largeness'' in group theory.
type: Item
instance of: scholarly article
title: On the concept of ``largeness'' in group theory.
zbMATH Open document ID: 1215.20035
author: Simon Thomas
published in: Journal of Algebra
publication date: 2010-03-04T00:00:00Z
review text: The article is a study of the Borel complexity of the following relations first introduced by \textit{S. J. Pride} [Word problems II, Stud. Logic Found. Math. Vol. 95, 299-335 (1980; Zbl 0438.20023)]: \(G_1\) is larger than \(G_2\) (\(G_1\succeq G_2\)) if there are finite-index subgroups \(H_i\leq G_i\) and a finite normal subgroup \(N_2\triangleleft H_2\) such that \(H_1\) surjects onto \(H_2/N_2\). This induces an equivalence relation \(G_1\approx G_2\): iff \(G_1\succeq G_2\) and \(G_2\succeq G_1\). The main results are Theorems 1.4 and 1.7, stating that on the set of finitely generated groups with fixed countable base, \(\approx\) (resp. the bi-surjectability equivalence relation) is a universal \(K_\sigma\) equivalence relation. -- The proofs mostly rely on combinatorial group theory.
reviewed by: Adrien Deloro
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 20F06
zbMATH DE Number: 5677308
zbMATH Keywords: Borel equivalence relations
zbMATH Keywords: combinatorial group theory
zbMATH Keywords: largeness of groups
zbMATH Keywords: finitely generated groups
zbMATH Keywords: subgroups of finite index
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.08.019
OpenAlex ID: W2014961654
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DOI: 10.1016/J.JALGEBRA.2009.08.019
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description: scientific article; zbMATH DE number 5677308